Elements of Passive Filter Design J. Sebek July 26, 2001 1 Introduction The design of arbitrary ¯lters is a complex subject about which much has been written [1][2][3]. It is not the purpose of this note to repeat all of that informa- tion. Rather, this note concentrates on the few elements that are required to design and synthesize some elementary ¯lters that are commonly used. With the information presented here, one should, using only some simple numerical calculations, be able to design and build these common ¯lters. This note will concentrate on the design of lowpass ¯lters. It will start with a discussion of several types of ¯lter designs, noting their strengths and weaknesses, and give the equations necessary to design the ¯lter transfer func- tions. From this viewpoint, the transfer functions describe the transmission of signals from the source through the network to the load. Next the synthesis of simple ladder circuits will be reviewed. This approach views the network and load as an impedance. Standard two port network theory gives the relationship between the transmission and impedance representations. The elements of this theory necessary to make this relation will be reviewed. Using this knowledge, the lowpass ¯lters can be synthesized as common ladder circuits. Finally the lowpass to bandpass transformation will be presented. This transformation can be used to design most band-pass ¯lters of interest. 2 Filter Types The purpose of a ¯lter is to condition an input signal in a way that preserves the signal characteristics of interest while eliminating the undesirable components. The characteristics can be speci¯ed in the frequency domain, the time domain, or a combination of both. However, ¯lter design is most easily accomplished in the frequency domain, so the ¯rst step in any of the designs discussed here will be to make the speci¯cation in that domain. 2.1 Transfer Function The variable of interest in the frequency domain is the radian frequency !. The frequency domain ¯lter speci¯cation, the transfer function, can be denoted 1 in several ways. Since this function a®ects both the amplitude and phase of the input signal as a function of frequency it is convenient to represent it in complex notation. To emphasize the complex nature of the function and the way in which the frequency enters into it, the transfer function is often denoted H (i!), where i = p 1. For notational purposes, a variable s = i! is often introduced so that th¡e transfer function can also be written as H (s). Transfer functions realizable with discrete components are polynomials. The complex conjugate of such a transfer function satis¯es the Schwarz re°ection principle [4] H¤ (i!) = H ( i!) : ¡ The complex transfer function can be separated into its amplitude and phase H (s) = H (s) eiÁ(s) j j When approximating the phase of H (s), one can work with the transfer function itself. But when dealing with the amplitude, it is often more convenient to work, not with the magnitude itself, but with the square of the magnitude, since this can be represented by a polynomial fraction. The square can be represented either as H (i!) 2 = H (i!) H ( i!) j j ¡ or H (s) 2 = H (s) H ( s) : j j ¡ In general, the ¯lter approximations are ideal; they cannot be represented by polynomials of ¯nite order. The approximations to the ideal improve with the complexity and number of components of the ¯lter. A lumped element ¯lter, the kind discussed in this note, in general requires one reactive component for each order of the polynomial in H (s). One of the choices in the ¯lter design process is to determine the level of ¯lter complexity required to provide the desired level of signal processing. In order for the transfer function to be realizable, all of its poles must lie in the left half complex plane, that is, Re s 0. Once the square of the mag- nitude function is determined from othefr cgo·nsiderations, the realizable transfer function will be determined by this condition on the location of the poles. 2.2 Butterworth Approximation 2.2.1 Butterworth Transfer Function Perhaps the most common ¯lter approximation type is the Butterworth ¯lter. Its characteristic is that its low-pass ¯lter is \maximally °at" at the origin. The perfect Butterworth ¯lter would be perfectly °at at the origin; that is, all 2 derivatives of H (s) with respect to s would vanish there. For a ¯nite Butter- worth approximation of order n, the best polynomial fraction approximation is given by 1 H (s) 2 = j j 1 + C ( s2)n ¡ 1 = 1 + C ( 1)n s2n ¡ 1 = (1) 1 + ( 1)n s2n ¡ where C is a constant that sets the frequency scale of the ¯lter. In the last line, the scaling constant has been incorporated into the frequency variable s. In all subsequent discussions, unless otherwise noted, such scaling constants will be assumed incorporated into s. If one needs to calculate a ¯lter function for a particular frequency, one can start with a normalized function such as the one above, and in the end replace s with the appropriately scaled value s=s0, where s0 is the scale of the physical problem of interest. An analytic solution for the pole locations of the Butterworth ¯lter can be easily found. The 2n possible locations are the solutions of the equation i¼ 1+n s = e 2n i¼ i¼ = e 2 e 2n and the realizable poles are the n in the left half of the complex plane. One sees that these poles uniformly populate the unit circle. If n is even, the poles are complex pairs centered about the real axis; if n is odd, there is one real pole on the negative real axis, and the rest form pairs centered around that axis. 2.2.2 Butterworth Characteristics Although ¯lters are generally characterized by their transfer functions in the frequency domain, their e®ect on signals may be best described in either the frequency of time domain. All lowpass ¯lters pass DC and, above some fre- quency, attenuate the signals with an attenuation that grows with frequency. For a polynomial ¯lter as described above, the high frequency behavior is deter- mined by the order of the ¯lter. The di®erence in the response of the di®erent ¯lter families occurs in the intermediate frequencies. The Butterworth ¯lter is the \vanilla" ¯lter. Its frequency response is monotonic; its attenuation increases with frequency. It passes signal pulses reasonably well and its step response has some overshoot, but not an excessive amount. A half-power (3 dB) point can easily be calculated for this ¯lter. From equation 1, this corner frequency is easily seen to be s = 1. The Butterworth design is often the default ¯lter of choice. In applications in which the system needs a sharper transition between the passband and stopband, or in which the system needs a more faithful response to a pulse, however, other ¯lter families have better performance. 3 2.3 Chebyshev Approximation 2.3.1 Chebyshev Transfer Function Another common desired ¯lter response is one with \equal ripple" throughout the ¯lter passband. The magnitude squared of a transfer function satisfying this requirement is 1 H (i!) 2 = (2) 2 2 j j 1 + ± Tn (!) where ± is a measure of the passband ripple and Tn (!) is the Chebyshev poly- nomial of order n. The Tn (!) are de¯ned by the equation cos (n arccos !) ! 1 T (!) = : (3) n cosh (n arccosh !) j!j · 1 ½ j j ¸ Straightforward algebraic manipulation results in the recursion relation T0 (!) = 1 T1 (!) = ! Tn+1 (!) = 2!Tn (!) Tn 1 (!) ¡ ¡ which demonstrates that these functions are, in fact, polynomials. The locations of the poles of the Chebyshev transfer function can also be analytically calculated. The poles of equation 2 satisfy i T (!) = : n §± Since s = i!, this can also be written s i T = : (4) n i §± ³ ´ Setting w = x + iy and s = i cosh w = i cosh (x + iy) = i cosh x cos y sinh x sin y (5) ¡ and substituting in equation 3, equation 4 can be rewritten as i cosh nx cos ny + i sinh nx sin ny = : §± 4 Equating the real and imaginary parts of the this equation gives the possible values of x and y for the poles 2k 1 ¼ y = ¡ n 2 1 1 x = arcsinh : n ± Substituting these values back into equation 5 gives the pole locations of the transfer function in the left half complex plane sk = ¾k + i!k 1 1 2k 1 ¼ 1 1 2k 1 ¼ = sinh arcsinh sin ¡ + i cosh arcsinh cos ¡ : ¡ n ± n 2 n ± n 2 µ ¶ µ ¶ µ ¶ µ ¶ Geometrically, these poles are seen to lie on an ellipse in the complex plane described by the equation 2 2 ¾k !k 2 1 1 + 2 1 1 = 1: sinh n arcsinh ± cosh n arcsinh ± ¡ ¢ ¡ ¢ 2.3.2 Chebyshev Characteristics The frequency response of a Chebyshev lowpass ¯lter ripples in the passband. These ripples have equal magnitude, hence the term \equal ripple". The last ripple of the ¯lter always points toward greater attenuation, so that the transi- tion between the passband and the stopband is sharper for the Chebyshev than for the Butterworth; the Chebyshev is a better approximation to a \brick wall" ¯lter. This sharp transition is responsible for increased ringing in the time domain responses of this ¯lter family, an often undesireable property. This ¯lter can also be described by a cuto® frequency. At the edge of the ripple band, the magnitude squared of the transfer function is 1 H (1) 2 = : j j 1 + ±2 Only if ± = 1 has the amplitude decreased by 3 dB at s = 1.